Problem 59 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(x_{1}, x_{2}, \ldots, x_{n}\) be n observations, and let \(\bar{x}\) be their arithmetic mean and \(\sigma^{2}\) be the variance. Statement-1: Variance of \(2 x_{1}, 2 x_{2}, \ldots, 2 x_{n}\) is \(4 \mathrm{o}^{2}\). Statement- \(\mathbf{2}\) : Arithmeticmean \(2 x_{1}, 2 x_{2}, \ldots, 2 x_{n}\) is \(4 \bar{x}\). \([2012]\) (a) Statement- 1 is false, Statement- 2 is true. (b) Statement- 1 is true, statement- 2 is true; statement- 2 is a correct explanation for Statement-1. (c) Statement-1 is true, statement- 2 is true; statement- 2 is not a correct explanation for Statement- \(1 .\) (d) Statement- 1 is true, statement- 2 is false.

Step-by-Step Solution

Answer

(d) Statement 1 is true, Statement 2 is false.

Step 1: Understanding Statement 1

Statement 1 claims that if we have data points \(2x_1, 2x_2, \ldots, 2x_n\), the variance of this set is \(4\sigma^2\). Recall that variance is affected by scaling by a factor \(k\) such that the new variance is \(k^2\) times the original variance, because variance measures average squared deviations from the mean. With \(k = 2\), the variance becomes \(4\sigma^2\), hence Statement 1 is true.

Step 2: Understanding Statement 2

Statement 2 suggests that the arithmetic mean of \(2x_1, 2x_2, \ldots, 2x_n\) is \(4\bar{x}\). Calculating the mean for \(2x_1, 2x_2, \ldots, 2x_n\) gives \(\frac{2}{n}(x_1 + x_2 + \ldots + x_n) = 2\bar{x}\). Therefore, the arithmetic mean is \(2\bar{x}\), not \(4\bar{x}\), so Statement 2 is false.

Step 3: Analyzing the Options

With Statement 1 being true and Statement 2 being false, the correct answer aligns with option (d), which states: Statement 1 is true, Statement 2 is false.

Key Concepts

Arithmetic Mean

The arithmetic mean is a fundamental concept in statistics, representing the average of a set of numbers. To calculate the mean of a collection of numbers, you simply add all the numbers together and then divide by the total count of the numbers. For example, if the numbers are denoted as \( x_1, x_2, \ldots, x_n \), the arithmetic mean \( \bar{x} \) is calculated as:
\[\bar{x} = \frac{x_1 + x_2 + \ldots + x_n }{n}\]This measure gives us an idea of what is considered typical or central in the data set.
Understanding how the mean changes with scaled data is important. For instance, if each data point is multiplied by 2, the new mean will be twice the original mean, i.e., \( 2\bar{x} \). This illustrates that the arithmetic mean is sensitive to uniform scaling of all data points.

Scaling Effect on Variance

Variance is a statistic that measures the dispersion of data points in a dataset. It gives us an idea of how much the data varies from the arithmetic mean. The larger the variance, the more spread out the data points are.
When scaling data by a constant factor, the effect on variance is not linear. Instead, if every data point in a dataset is multiplied by a factor \( k \), the variance of the new dataset becomes \( k^2 \) times the original variance \( \sigma^2 \). This is because variance involves averaging the squared deviations.
Thus, if each data element is multiplied by 2, the variance of this new dataset becomes \( 4\sigma^2 \), since \( (2)^2 = 4 \). This scaling principle is an essential concept in understanding data transformations and their effects.

JEE Main Mathematics

JEE Main Mathematics is an entrance exam for engineering colleges in India that tests a range of mathematical topics. It requires a deep understanding of various concepts from algebra to calculus, including statistics.
Within the domain of statistics, questions often explore understanding of basic concepts like arithmetic mean and variance, as well as their properties and transformations.
For students preparing for the exam, it's crucial to grasp the foundational principles behind each formula and operation. For example, knowledge about how variance changes with scaled data and accurate calculations of the arithmetic mean are frequently tested.

• Being accurate and efficient in solving such problems can give a significant advantage in the exam.
• Practice with different problem scenarios helps build confidence and proficiency.

Statistical Concepts

Statistical concepts are the backbone of analyzing and interpreting data in meaningful ways. Two key statistical metrics are the arithmetic mean and variance. Each provides different insights into the data: the mean gives a central tendency, while variance provides information on data spread.
Statistics involves understanding the behavior of these metrics under various operations, such as scaling or addition of constants.

• The arithmetic mean of scaled data is simply the scale factor times the original mean.
• The variance of scaled data involves squaring the scale factor, thus highlighting its sensitivity to changes in magnitude.

These foundational ideas are crucial not only in academics but in real-world data science and analytics, where data often needs to be transformed for modeling and interpretation.